The BBC's Akwasi Sarpong in the capital, Lusaka, say there was a high turnout at polling centres in the city.

Long, calm, orderly queues formed early in the morning and election monitors have not reported any complaints, he says.

There were five votes - for president, MPs, mayors, local councillors and an amendment to the constitution on changes to the bill of rights.

As the counting begins, Zambia's electoral commission has urged political party supporters to remain calm after final results are announced.

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Police broke up skirmishes between rival political supporters in Lusaka just hours before polls opened

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A young lollipop seller in Lusaka hopes to make money from thirsty voters

Observers say Zambia's struggling economy will be a key issue.

Plunging prices for copper, its main export, have closed mines and left thousands unemployed. With economic growth roughly halved, the country asked the International Monetary Fund for help earlier this year.

In addition, Zambia, like other parts of southern Africa, has been hit by a drought that the UN has described as the worst in 35 years.

The UPND (United Party for National Development) has accused President Lungu of presiding over the "collapse" of the economy. But the PF (Patriotic Front) says it has a plan to diversify the economy.

Nail polish allowed

During the last election, some women wearing nail varnish were forced to remove it before voting as polling officials said they would not be able to apply the indelible ink correctly.

But on Wednesday night, the electoral commission circulated posts on social media saying women with "painted nails and/or false nails" could vote.

Zambia's fluid politics by Nomsa Maseko, BBC News, Lusaka

This is an election like no other. Each voter was given five ballot papers - presidential, parliamentary, mayoral, local government and a referendum. All of these to be decided on one day. The five ballot papers each represent a separate vote. They are orange, red, purple, black and tan.

The political landscape is fluid. There's been a shifting of political allegiances which in other parts of the continent would be seen as a betrayal.

Geoffrey Bwalya Mwamba is Hakainde Hichilema's running mate for the United Party for National Development. But before this, he was defence minister in late Michael Sata's Patriotic Front government.

While former Deputy President Guy Scott, whose wife is seeking a position as a lawmaker in Lusaka central constituency to replace her husband, has endorsed Edgar Lungu's main rival Hakainde Hichilema.

And lastly, Mulenga Sata, son of the late President Sata has also defected to Mr Hichilema's UPND.

Front-runners

Edgar Lungu - ruling Patriotic Front (PF)

Image copyrightAFP

The 59-year-old lawyer became president in January 2015 in an election called after the death in office of President Michael Sata

He served in Mr Sata's government as minister for justice and defence

Support base: Home area of Eastern province, the capital, Lusaka, and the Copperbelt plus the Bemba-speaking regions

Known for ordering a national day of prayer last year to help combat the economic problems facing the country

Married with six children, this vote is seen as a personal referendum on him

Hakainde Hichilema - opposition United Party for National Development (UPND)

Image copyrightAFP

The 54-year-old economist has contested nearly all elections in Zambia since 2006

A wealthy businessman, he has a solid track record in the private sector

Support base: Home region of Southern Province and is popular with younger voters

Known for being Zambia's second largest cattle rancher

Commonly referred to by his initials HH, he is married with three children

A watchdog warned on Wednesday that clashes between rival political groups over recent weeks could keep some voters away.

"Escalating levels of violence may have a negative impact on the elections and reduce voter turnout," the Zambian Elections Information Centre said in a statement.

"Political cadres have increasingly become unruly to the extent that they have shown no regard for law enforcement agents."

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There were five different votes

On Tuesday, the head of the electoral commission, Esau Chulu, warned the two front-runners to avoid stirring unrest.

"I do not think that either of you will want to go on record as having been the two political parties who contributed to permanently denting Zambia's record of peaceful elections," he said.

Campaigning was suspended for 10 days in Lusaka last month after a UPND supporter was shot dead during a protest.